Author(s)

M. Y. Melnikov

Abstract

An iterative semi-analytic procedure is developed for solution of problems arising in the pricing of American options. Introduction of a penalty function reduces the problem to a European options problem with a nonlinear term in the Black-Scholes equation. The approach is based on the use of a Green’s function constructed for a terminal-boundary value problem stated for the linear Black-Scholes equation. Different boundary conditions can potentially be treated within this approach. Closed analytic form of Green’s functions are obtained by a combination of the methods of Laplace transform and variation of parameters. A numerical experiment reveals high accuracy level attained in computing of solutions of linear problems that arise at each stage of the iterative procedure. Keywords: American options, Green’s function, iterative approach. 1 Introduction Upon implementing a penalty function approach in the way suggested in [1] and later used in [2], the pricing of American options can be simulated by the following nonlinear terminal-boundary value problem ∂v(S, t) ∂t + σ2S2 2 ∂2v(S, t) ∂S2 + rS ∂v(S, t) ∂S − rv(S, t) + εC v(S, t) + ε − q(S) = 0 (1) v(S, T ) = ϕ(S) (2) v(S1, t) = A(t) and v(S2, t) = B(t) (3) posed for the two variable function v(S, t) in the rectangular region  = (S1 < S < S2) × (0 < t < T) of the S, t-plane.

Keywords

American options, Green’s function, iterative approach.